Добавил:
ivanov666
Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз:
Предмет:
Файл:Функции Грина и проводимость. Учебное пособие
.pdf
ଶ
ɢ ɫɪɚɡɭ ɨɩɭɫɤɚɟɦ ɜɟɥɢɱɢɧɵ, ɤɨɬɨɪɵɟ ɢɦɟɸɬ ɩɨɪɹɞɨɤ ߜ
ଶ
, ߟߞ
ɭɫɪɟɞɧɟɧɢɢ ɧɭɥɶ: ߞ
݊ሺെ݅ȟɁሻܾ
ɢ ɬ. ɩ. ɉɪɨɫɬɨɟ ɜɵɱɢɫɥɟɧɢɟ ɞɚɟɬ
ܾ
ିଵ
ሺ
ሺݐሻെ݊ሺ
ݐߜሻെܾ
݅ȟ
ሺݐሻ
ൌെʹ݊߱
כ
ାଵ
ߜሻܾ
ሺݐሻ
െ݊
௦
ߜܾ
ଶ
ɢ ɤɨɬɨɪɵɟ ɞɚɞɭɬ ɩɪɢ
ሺݐሻ
ଶ
ଶ
ሺ
ʹȁߞ
ሻ
ȁ
ߟ
ܾ
ሺݐሻǤሺ
ͳʹǤʹͷ
ɍɫɪɟɞɧɢɦ ɷɬɨ ɭɪɚɜɧɟɧɢɟ, ɢ ɩɪɢ ɷɬɨɦ ɭɱɬɟɦ ɨɩɪɟɞɟɥɟɧɢɟ (12.16), ɤɨɪɪɟɥɹɬɨɪɵ
(12.26), ɚ ɬɚɤɠɟ ɬɨ, ɱɬɨ ɜɟɥɢɱɢɧɵ ܤ
Ͳൌെʹ݊߱௦െ݊ଶ൬
ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɤɨɨɪɞɢɧɚɬɵ ݐ
ʹ
ͳ
߬
߬
ଵ
൰൨ܤെ݅݊ȟܤ
ଶ
ିଵ
െ݅݊ȟכܤ
ାଵ
ሺ
Ǥ
ͳʹǤʹ
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɹ
ʹ
ͳ
ൌ
߬
ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ
ሺ݊ߚ
ሺ
ͳʹǤʹሻɩɪɢɦɟɬ ɜɢɞ
ሻ
ܤ
݅ɀܤ
௦
Ⱥɧɚɥɨɝɢɱɧɨɟ ɜɵɱɢɫɥɟɧɢɟ, ɩɪɨɞɟɥɚɧɧɨɟ ɞɥɹ ɜɟɥɢɱɢɧɵ ܽ
ͳ
߬
ǡߚ௦ൌʹ߱௦߬ǡߛൌȟ߬ǡ
߬
ଵ
ଶ
݅ߛכܤ
ିଵ
ൌͲǡሺ݊ൌͲǡͳǡʹǥሻǤ
ାଵ
ሺ
݅߱
ǡݐሻ, ɞɚɟɬ ɞɥɹ ܣ
௦
ሺ
ͳʹǤʹ
ሺ
ͳʹǤʹͺ
ɫɥɟɞɭɸɳɟɟ ɭɪɚɜɧɟɧɢɟ
ሺ݊ߚ
ሻ
ܣ
െ݅ɀܣ
௦
ାଵ
െ݅ߛכܣ
ൌͲǡሺ݊ൌͲǡͳǡʹǥሻǤ
ିଵ
ሺ
ͳʹǤʹͻ
ɉɪɟɠɞɟ ɱɟɦ ɢɫɤɚɬɶ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦ (12.28), (12.29), ɩɪɨɜɟɞɟɦ ɭɫɪɟɞɧɟɧɢɟ ɡɚɩɚɡɞɵɜɚɸɳɟɣ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢɦɟɧɢɦ ɮɨɪɦɭɥɭ (8.69), ɩɨɥɨɠɢɜ
ɜ ɧɟɣ ݐൌݐ
ᇱ
݃ො
ఠԜଵଶ
ሺ
ݐǡݐሻൌെ
መ
ܵ
ఠԜଵଶ
ሺ
λǡݐሻܵ
መ
ܵ
ఠԜଵଵ
መ
ఠԜଵଵ
ሺ
λǡെλ
ሺ
ݐǡെλሻԜ
ሻ
ǡ
ሺ
ͳʹǤ͵Ͳ
ɞɚɥɟɟ ɢɡ ɝɪɭɩɩɨɜɨɝɨ ɫɜɨɣɫɬɜɚ ܵ-ɦɚɬɪɢɰɵ ɢɦɟɟɦ
ܵ
መ
ሺ
λǡെλሻൌܵ
ଵଵ
መ
ଵଵ
ሺ
λǡݐሻܵ
መ
ሺ
ݐǡെλሻܵ
ଵଵ
መ
ଵଶ
ሺ
λǡݐሻܵ
መ
ଶଵ
ሺ
ݐǡെλሻǡ
ሺ
ͳʹǤ͵ͳ
ɨɬɤɭɞɚ
ൣܵ
መ
ሺ
ଵଵ
λǡെλሻ൧
ିଵ
ൌሺെͳ
ஶ
ୀ
ሺ
λǡݐሻܵ
λǡݐሻܵ
መ
ଶଵ
መ
ଵଵ
ሺ
ݐǡെλሻ൧
ሺ
ݐǡെλሻ൧
መ
ൣܵ
ଵଶ
ሻ
መ
ሺ
ൣܵ
ଵଵ
ାଵ
ሺ
ǡ
ͳʹǤ͵ʹ
ɩɨɞɫɬɚɧɨɜɤɚ ɜ (12.42)
ஶ
݃ො
ሺ
݅߱
ǡݐǡݐሻൌെሺെͳ
௦
ଵଶ
ୀ
ାଵ
ሻ
ሺ
݅߱
ܿ
ǡݐሻ݀
௦
ାଵ
ሺ
݅߱
௦
ǡݐሻǡ
ሺ
ͳʹǤ͵͵
ɝɞɟ
ܿሺ݅߱
ǡݐሻൌ
௦
መ
ܵ
ఠೞଵଶ
መ
ܵ
ఠೞଵଵ
ሺ
ሻ
λǡݐ
ǡ݀ሺ݅߱௦ǡݐሻൌ
ሺ
ሻ
λǡݐ
መ
ܵ
ఠೞଶଵ
መ
ܵ
ఠೞଵଵ
ሺ
ݐǡെλ
ሺ
ݐǡെλ
ሻ
Ǥ
ሻ
ሺ
ͳʹǤ͵Ͷ
131
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ɍɫɪɟɞɧɟɧɢɟ (12.45) ɞɚɟɬ
ஶ
ሺ
ሻ
݅߱
ۄ
௦
ൌെሺെͳ
ୀ
ۃ
݃ො
ଵଶ
ሺ
ሻ
ሺ
ሻ
ܥ
ାଵ
݅߱
ܦ
௦
ሻ
݅߱
ǡ
௦
ሺ
ͳʹǤ͵ͷ
ɝɞɟ
ܥ
௦
ሻ
ൌۃܿ
ሺ
݅߱
ሻ
ۄ
ǡݐ
௦
ǡܦ
ሺ
݅߱
௦
ሺ
݅߱
ሻ
ൌۃ݀
ሺ
݅߱
ሻ
ۄ
ǡݐ
௦
ሺ
ͳʹǤ͵
ɞɚɥɶɧɟɣɲɟɟ ɪɚɫɫɦɨɬɪɟɧɢɟ ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ, ɡɚɦɟɬɢɜ ɫɥɟɞɭɸɳɟɟ. ɂɡ (8.51) ɢ
(8.54) ɦɨɠɧɨ ɭɜɢɞɟɬɶ ɫɥɟɞɭɸɳɢɟ ɫɜɨɣɫɬɜɚ ܵ-ɦɚɬɪɢɰɵ
ܵ
ܵ
መ
ఠԜଵଶ
መ
ఠԜଵଵ
ᇱ
ሺ
ݐǡݐ
ǢሼߟǡߞǡߞכǢȟǡȟ
ᇱ
ሺ
ݐǡݐ
ǢሼߟǡߞǡߞכǢȟǡȟ
ɚ ɩɨɫɤɨɥɶɤɭ ɫɪɟɞɧɢɟ ɢɧɜɚɪɢɚɧɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ߞ֎ߞ
כ
መ
ሻ
ሽ
ൌܵ
ିఠԜଶଵ
כ
መ
ሻ
ሽ
ൌܵ
ିఠԜଶଶ
ᇱ
ሺ
ݐǡݐ
ǢሼെߟǡߞכǡߞǢെȟכǡെȟ
ᇱ
ሺ
ݐǡݐ
ǢሼെߟǡߞכǡߞǢെȟכǡെȟ
ሻǡሺ
ሽ
ሻǡሺ
ሽ
כ
, ߟെߟ,
ͳʹǤ͵
ͳʹǤ͵ͺ
ɡɚɤɥɸɱɚɟɦ
ǡȟǡȟ
௦
ǡȟǡȟ
௦
כ
כ
ሻ
ሻ
ൌܣ
ൌܤ
ሺ
݅߱
ሺ
݅߱
ǡെȟכǡെȟሻǡ
௦
ǡെȟכǡെȟሻǤ
௦
ሺ
ͳʹǤ͵ͻ
ሺ
ͳʹǤͶͲ
ሺ
ܥ
െ݅߱
ሺ
െ݅߱
ܦ
ɋɨɩɨɫɬɚɜɥɹɹ ɷɬɢ ɪɚɜɟɧɫɬɜɚ ɫ ɪɟɲɟɧɢɹɦɢ (12.44), ɦɵ ɩɨɥɭɱɚɟɦ
ሺ
ሻ
ሺ
െ݅߱
ܥ
௦
ൌܤ
݅߱
ሻ
ǡܦ
௦
ሺ
െ݅߱
௦
ሻ
ൌܣ
ሺ
ሻ
݅߱
ǡ
௦
ሺ
ͳʹǤͶͳ
ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ
ۃ݃ො
ଵଶ
ሺ
െ݅߱
ሻ
ۄ
௦
ൌۃ݃ො
ଵଶ
ோ
ሺ
݅߱
ሻ
ۄ
ǡ
௦
ሺ
ͳʹǤͶʹ
ɩɨɷɬɨɦɭ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (12.10) ɩɪɢɦɟɬ ɜɢɞ:
ଶ
ൌെ
ʹ݅ܶ
ݒ
ۃ݃ො
ఠೞவ
ଵଶ
ோ
ሺ
݅߱
ǡݖȀݒǡݖȀݒ
௦
ሻ
ۄ
Ǥ
ሺ
ͳʹǤͶ͵
ȟ
݃
12.1.3. ȼɵɪɚɠɟɧɢɟ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ ɱɟɪɟɡ ɧɟɩɪɟɪɵɜɧɭɸ ɞɪɨɛɶ
Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦ (12.28), (12.29) ɫ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ܣ
ൌܤଵൌͳɦɨɠɧɨ
ଵ
ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ܣ
ɝɞɟ ܳ
ɟɫɬɶ ɧɟɩɪɟɪɵɜɧɚɹ ɞɪɨɛɶ
ܳ
ሺ
ߚ
௦
ൌሺ݅ߛ
ଶ
ȁ
ǡȁߛ
ሻ
כሻ
ൌ
ෑܳ
ୀଵ
݊ߚ
ǡ ܤൌሺെ݅ߛ
ͳ
௦
݊ͳߚ௦
ȁߛȁ
ሻ
ෑܳ
ǡ
ୀଵ
ଶ
ଶ
ȁߛȁ
݊ʹߚ௦ڮ
ሺ
ͳʹǤͶͶ
ሺ
Ǥ
ͳʹǤͶͷ
132
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ȼ ɷɬɨɦ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ. ȼɨ-ɩɟɪɜɵɯ, ɢɡ (12.45) ɨɱɟɜɢɞɧɨ
ͳ
ൌ݊ߚ௦ȁߛ
ܳ
ଶ
ȁ
ܳ
ǡ
ାଵ
ሺ
ͳʹǤͶ
ɜɨ-ɜɬɨɪɵɯ, ɩɨɞɫɬɚɜɢɜ ɜɵɪɚɠɟɧɢɹ ɢɡ (12.44) ɜ (12.28) ɢ (12.29), ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ (12.27). ɉɨɞɫɬɚɜɢɦ ɬɟɩɟɪɶ ɪɟɲɟɧɢɹ (12.35) ɜ (12.17), ɩɨɥɭɱɢɦ
ோ
ሺ
ଵଶ
݅߱
ሻ
ۄۄ
௦
ۃۃ
݃ො
ൌ݅ߛሺܳ
ஶ
ൌ݅ߛሺെȁߛ
ୀ
ଶ
ሺ
ଵ
ሻ
ȁ
ߚ
௦
ǡȁߛ
െȁߛ
ȁ
ଶሻ
ȁ
ቌෑܳ
ଶ
ଶ
ܳ
ଵ
ୀଵ
ሺ
ଶ
ȁ
ߚ
ǡȁߛ
௦
ଶ
ሺ
ଶ
ሻ
ȁ
ߚ
ሻ
ቍܳ
ǡȁߛ
௦
ሺ
ܳ
ߚ
ଶ
ାଵ
ଶ
ሻ
ȁ
ڮሻǤ
ǡȁߛ
௦
ଶ
ሺ
ሻ
ȁ
ߚ
௦
ǡȁߛ
ൌ
ሺ
ͳʹǤͶ
ɑɬɨɛɵ ɧɚɣɬɢ ɜɵɪɚɠɟɧɢɟ ɷɬɨɝɨ ɪɹɞɚ, ɡɚɦɟɬɢɦ, ɱɬɨ ɢɡ (12.45) ɥɟɝɤɨ ɭɜɢɞɟɬɶ
ɫɨɨɬɧɨɲɟɧɢɟ, ɜɵɬɟɤɚɸɳɟɟ ɢɡ ɩɪɚɜɢɥɚ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɞɪɨɛɢ:
߲ܳ
ሺ
ሻ
ߚ
ǡݔ
௦
߲ݔ
ൌെܳ
ଶ
ሺ
ߚ
ǡݔሻܳ
௦
ାଵ
ሺ
ߚ
ǡݔሻെݔܳ
௦
ଶ
ሺ
ሻ
ߚ
ǡݔ
௦
߲ܳ
ାଵ
߲ݔ
ሺ
ሻ
ߚ
ǡݔ
௦
ǡሺͳʹǤͶͺ
ɢɬɟɪɢɪɭɹ ɷɬɨ ɪɚɜɟɧɫɬɜɨ, ɧɚɱɢɧɚɹ ɫ ݊ൌͳ ɩɨɥɭɱɢɦ
ሺ
ሻ
ߚ
ǡݔ
߲ܳ
ଵ
௦
߲ݔ
ൌെܳ
ଶ
ሺ
ߚ
ଵ
ǡݔሻܳ
௦
ሺ
ߚ
ଶ
௦
ǡݔሻݔܳ
ଶ
ሺ
ଵ
ߚ
ǡݔሻܳ
௦
ଶ
ሺ
ߚ
ଶ
ǡݔሻܳ
௦
ሺ
ߚ
ǡݔሻڮ ǤሺͳʹǤͶͻ
ଷ
௦
ɫɪɚɜɧɢɜɚɹ ɷɬɨ ɫ (12.47), ɩɨɥɭɱɚɟɦ
ሺ
ሻ
ߚ
ǡݔ
ଵ
௦
߲ݔ
Ǥ
௫ୀȁఊ
మ
ȁ
൱ൌ
ሺ
ͳʹǤͷͲ
ǡݔሻ൨
ȁ
ଶ
ቈ
௫ୀȁఊ
߲ܳ
మ
ȁ
ோ
ሺ
ሻ
݅߱
ۄۄ
௦
ൌ݅ߛ൭ܳ
ଵ
ۃۃ
݃ො
ଵଶ
ൌ݅ߛ
ሺ
ߚ
߲
߲ݔ
௦
ǡȁߛ
ݔܳ
ଶ
ሻ
ȁ
ȁߛ
ሺ
ߚ
ଵ
௦
ɍɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (12.43) ɫ ɭɱɟɬɨɦ (12.50) ɩɪɢɦɟɬ ɜɢɞ
ʹܶ߬
ͳ
ൌ
ଶ
݃
ݒ
ఠೞவ
߲
߲ݔ
ݔܳ
ሺ
ߚ
ǡݔሻ൨
ଵ
௦
௫ୀȁఊ
ǡ
మ
ȁ
ሺ
ͳʹǤͷͳ
ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ȁȟȁ ɨɬ ܶ ɢ ߬.
12.1.4. ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɧɟɩɪɟɪɵɜɧɨɣ ɞɪɨɛɢ ɤɚɤ ɨɬɧɨɲɟɧɢɟ
ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ
Ɉɤɚɡɵɜɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɚɹ ɞɪɨɛɶ ܳ
ଶ
ሺ
ሻ
ȁ
ߚ
ɦɨɠɟɬ ɛɵɬɶ ɜɵɪɚɠɟɧɚ ɫ ɩɨɦɨ-
ǡȁߛ
௦
ɳɶɸ ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ ɦɧɢɦɨɝɨ ɚɪɝɭɦɟɧɬɚ. Ɇɵ ɢɫɩɨɥɶɡɭɟɦ ɪɟɤɭɪɫɢɜɧɵɟ ɮɨɪɦɭɥɵ Ƚɚɭɫɫɚ ɞɥɹ ɝɢɩɟɪɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɮɭɧɤɰɢɹ
ܿሺܿͳሻܨሺܽǡܾǡܿǢݖሻെܿሺܿͳሻܨሺܽǡܾͳǡܿͳǢݖሻ
ܽሺܿെܾሻݖܨሺܽͳǡܾͳǡܿʹǢݖሻൌͲǡ
ሺ
ͳʹǤͷʹ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
133
ሺ
ܿͳሻܨሺܽǡܾǡܿǢݖሻെܿሺܿͳሻܨሺܽͳǡܾǡܿͳǢݖሻ
ܾሺܿെܽሻݖܨሺܽͳǡܾͳǡܿʹǢݖሻൌͲǤ
ሺ
ͳʹǤͷ͵
ሻ
ɉɨɞɟɥɢɦ ɩɟɪɜɨɟ ɪɚɜɟɧɫɬɜɨ ɧɚ ܿሺܿͳሻܨሺܽǡܾͳǡܿͳǢݖ
ܨሺܽǡܾǡܿǢݖ
ሻ
ܨሺܽǡܾͳǡܿͳǢݖ
ൌͳെ
ሻ
ሾ
ܽሺܿെܾሻȀܿሺܿͳ
ܨሺܽǡܾͳǡܿͳǢݖ
ܨሺܽͳǡܾͳǡܿʹǢݖ
ൌͳെ
ሻ
ሻ
ܾሺܿെܽ
ܿሺܿͳ
ܨሺܽͳǡܾǡܿͳǢݖ
ɚ ɜɬɨɪɨɟ ɧɚ ܿሺܿͳሻܨሺܽͳǡܾǡܿͳǢݖ
ܨሺܽǡܾǡܿǢݖ
ܨሺܽͳǡܾǡܿͳǢݖ
ሻ
ܨሺܽͳǡܾͳǡܿʹǢݖ
ሻ
:
ሻ
ሿ
ݖ
ሻ
ሺ
ǡ
ͳʹǤͷͶ
ሻ
ሻ
൨ݖ
ሻ
ሻ
ሺ
Ǥ
ͳʹǤͷͷ
ሻ
ȼ ɪɚɜɟɧɫɬɜɟ (12.59) ɫɞɟɥɚɟɦ ɡɚɦɟɧɭ ܾ՜ܾͳ ɢ ܿ՜ܿͳ ɢ ɩɨɞɫɬɚɜɢɦ ɜ
(12.58) ɢ ɩɨɜɬɨɪɢɦ ɩɪɨɰɟɫɫ ɰɢɤɥɢɱɟɫɤɢ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɧɟɩɪɟɪɵɜɧɭɸ
ɞɪɨɛɶ
ܨሺܽǡܾͳǡܿͳǢݖ
ܨሺܽǡܾǡܿǢݖ
ሻ
ɇɟɩɪɟɪɵɜɧɭɸ ɞɪɨɛɶ ܳ
ܳ
ሺ
ߚ
ଵ
௦
ሻ
ǡȁߛ
ൌ
ͳെ
ͳെ
ͳെ
ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ
ଵ
ൌ൬
ͳ
ߚ
ͳ
௦
ଶ
ሻ
ȁ
ሾ
ሺ
ሾ
ܾͳሻሺܿͳെܽሻȀሺܿͳሻሺܿʹ
ሺ
ሾ
ܽͳሻሺܿͳെܾሻȀሺܿʹሻሺܿ͵
൰
ͳ
ͳ
ܽሺܿെܾሻȀܿሺܿͳ
ͳڮ
ͳ
ଶ
ȁߛȁ
Ȁሺߚ௦ͳሻሺߚ௦ʹ
ଶ
ȁߛȁ
ͳ
Ȁሺߚ௦ʹሻሺߚ௦͵
ͳڮ
ሻ
ሿ
ݖ
ሻ
ǤሺͳʹǤͷ
ሻ
ሿ
ݖ
ሻ
ሿ
ݖ
ሺ
ǡ
ͳʹǤͷ
ሻ
cɪɚɜɧɢɜɚɹ ɫ (12.56), ɦɵ ɩɨɥɭɱɢɦ
ଶ
ሻ
ܳ
ଶ
ሺ
ȁ
ߚ
ǡȁߛ
ଵ
௦
ͳ
ሻ
ൌ
ߚ
ͳ
௦
ܨሺܽǡܾͳǡߚ௦ʹǢȁߛ
ǡ՜ஶ
ܨሺܽǡܾǡߚ
ͳǢȁߛ
௦
ȁ
Ȁܾܽ
ଶ
ሻ
ȁ
Ȁܾܽ
ሺ
ǡ
ͳʹǤͷͺ
ሻ
ሻ
ሻ
ሻ
ሻ
Ⱦɥɹ ɩɪɟɞɟɥɚ ɢɦɟɟɦ
ǡ՜ஶ
ൌȞሺߚ
ܨሺܽǡܾǡߚ௦ͳǢȁߛ
ͳሻቈ
௦
Ȟሺߚ
௦
ൌȞሺߚ
ଶ
ȁ
Ȁܾܽሻൌͳ
ͳ
ሻ
ͳ
ͳ
௦
ଶ
ȁߛȁ
Ȟሺߚ௦ͳ
ିఉ
ೞ
ሻ
ȁߛȁ
ܫ
134
మ
ȁఊȁ
ఉೞାଵ
ሻ
Ȟሺߚ௦͵ሻʹǨ
ሺ
ʹȁߛ
ఉ
ೞ
ሺ
ఉ
ାଵሻሺఉೞାଵሻଶǨ
ೞ
ȁߛȁ
ሻ
ȁ
ǡ
ȁఊȁ
ସ
ర
ڮൌ
ڮൌ
ሺ
ͳʹǤͷͻ
ሻ
ɩɨɞɫɬɚɜɥɹɹ ɜ (12.58), ɩɨɥɭɱɢɦ
ሺ
ሻ
ȁ
ʹȁߛ
ܫ
ͳ
ଶ
ሺ
ܳ
ଵ
ሻ
ȁ
ߚ
ǡȁߛ
௦
ఉೞାଵ
ൌ
ȁߛȁ
ሺ
ܫ
ఉ
ೞ
ʹȁߛ
ǡ
ሻ
ȁ
ሺ
ͳʹǤͲ
ɭɱɬɟɦ ɫɨɨɬɧɨɲɟɧɢɟ
ܫ
ఔାଵ
ሺݖሻ
ൌܫ
ఔ
ሺݖሻ
ሺݖሻ
െ
ǡ
ܫ
ఔ
ݖ
ሺ
ͳʹǤͳ
ߥ
ᇱ
ɬɨɝɞɚ ɥɟɝɤɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɢɦɟɟɬ ɦɟɫɬɨ
ͳ
߲
߲ݔ
ݔܳ
ሺ
ߚ
ǡݔሻ൨
ଵ
௦
௫ୀȁఊ
మ
ȁ
ൌʹ
߲
߲
ݖ
ݖ
߲ݖ
߲ݖ
ܫ
ሺݖሻ
൨
௭ୀଶȁఊ
ǡ
ȁ
ఉ
ೞ
ሺ
ͳʹǤʹ
ɜ ɢɬɨɝɟ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (12.51)) ɩɪɢɦɟɬ ɜɢɞ
ͳ
߲
ʹɎݒ
ൌͶߨܶ߬
ଶ
݃
ఠ
߲
ݖ
ݖ
߲ݖ
߲ݖ
வ
ೞ
ܫ
ሺݖሻ
൨
௭ୀଶȁఊ
Ǥ
ȁ
ఉ
ೞ
ሺ
ͳʹǤ͵
12.1.5. Ɍɟɦɩɟɪɚɬɭɪɚ ɩɟɪɟɯɨɞɚ
ɉɟɪɟɣɞɟɦ ɤ ɨɩɪɟɞɟɥɟɧɢɸ ɤɪɢɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɩɟɪɟɯɨɞɚ ɜ ɩɪɢɫɭɬɫɬɜɢɢ
ɩɪɢɦɟɫɟɣ. ɗɬɨɦɭ ɫɥɭɱɚɸ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɟɞɟɥ
ȁߛȁ
՜Ͳ, ɩɨɷɬɨɦɭ ɢɫɩɨɥɶɡɭɟɦ
ɚɫɢɦɩɬɨɬɢɤɭ ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɨɣ ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ ([14], ɫɬɪ. 196):
ܫ
ఉ
ೞ
ൌቀ
ݖ
ቁ
ʹ
ሺݖሻ
ఉ
ೞ
Ȟሺߚ
ͳ
௦
ͳ
ሻ
ቈͳ
ሺ
ݖȀʹ
ߚ௦ͳ
ଶ
ሻ
ʹሺߚ௦ͳሻሺߚ௦ʹ
ሺ
ݖȀʹ
ସ
ሻ
ሻ
ሺݖ
ସ
ሻǡሺ
ͳʹǤͶ
ɨɬɤɭɞɚ
ܫ
ଶ
ሺ
ሻ
ሺݖሻ
ൌߚ
ఉ
ೞ
ݖ
௦
ݖȀʹ
ߚ௦ͳ
െ
ʹሺߚ௦ͳ
ሺ
ݖȀʹ
ሻ
ସ
ሻ
ଶ
ሺ
ߚ
௦
ʹ
ǡ
ሻ
ሺ
ͳʹǤͷ
ɢ ɧɚɤɨɧɟɰ ɜ ɷɬɨɦ ɩɪɢɛɥɢɠɟɧɢɢ
ଶ
ͳ
߲
߲
ݖ
ݖ
߲ݖ
߲ݖ
ܫ
ሺݖሻ
൨
ఉ
ೞ
௭ୀଶȁఊ
ͳ
ൌ
ȁ
െ
ͳ
ሺ
ߚ
௦
ߚ
௦
ͳ
ʹȁߛ
ሻ
ȁ
ଶ
ሺ
ߚ
ʹ
௦
ሺ
Ǥ
ሻ
ͳʹǤ
ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɩɨɬɪɟɛɭɟɬɫɹ ɬɨɥɶɤɨ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɩɨɞɫɬɚɜɥɹɹ ɟɝɨ ɜ
(12.63), ɩɨɥɭɱɢɦ
ஶ
ൌ
ݏͳȀʹͳȀͶߨܶ
௦ୀ
ͳ
Ǥ
߬
ሺ
ͳʹǤ
ʹɎݒ
ൌͶߨܶ߬
ଶ
݃
ఠೞவ
ͳ
ߚ
ͳ
௦
Ɏɨɪɦɚɥɶɧɨ ɫɭɦɦɚ ɜ (12.67) ɪɚɫɯɨɞɢɬɫɹ, ɨɞɧɚɤɨ ɷɬɨ ɪɚɫɯɨɞɢɦɨɫɬɶ ɮɢɤɬɢɜɧɚɹ,
ɨɧɚ ɫɜɹɡɚɧɚ ɫ ɬɟɦ, ɱɬɨ ɮɭɧɤɰɢɢ Ƚɪɢɧɚ, ɤɨɬɨɪɵɦɢ ɦɵ ɩɨɥɶɡɨɜɚɥɢɫɶ, ɜɟɪɧɵ ɬɨɥɶɤɨ
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɮɟɪɦɢ-ɩɨɜɟɪɯɧɨɫɬɢ, ɤɨɬɨɪɚɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɬɨ-
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
135
ɢ െ. ȼɟɥɢɱɢɧɚ ɷɬɨɣ ɨɤɪɟɫɬɧɨɫɬɢ ɩɨɪɹɞɤɚ ߝ
ɱɟɤ
ɧɭɠɧɨ ɨɝɪɚɧɢɱɢɬɶ ɦɚɤɫɢɦɚɥɶɧɵɦ ɢɧɞɟɤɫɨɦ ݏ
, ɩɨɷɬɨɦɭ ɫɭɦɦɢɪɨɜɚɧɢɟ
ி
௫ߝி
Ȁʹߨܶ. Ɇɨɠɧɨ, ɨɞɧɚɤɨ,
ɨɛɨɣɬɢ ɷɬɭ ɩɪɨɛɥɟɦɭ, ɞɥɹ ɷɬɨɝɨ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ ɞɥɹ ɱɢɫɬɨɝɨ ɦɟɬɚɥɥɚ (11.34)
ൌ൬
ʹߛߝ
ߨܶ
ி
൰ǡ
ሺ
ͳʹǤͺ
ʹɎݒ
݃
ଶ
ሻ
ɚ ɤɪɨɦɟ ɬɨɝɨ, ɭɠɟ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ܶاߝ
ʹߛߝ
ி
ߨܶ
൰ൌ
൬
ɢɦɟɟɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɨ
ி
ఌಷȀଶగ்
௦ୀ
ͳ
ݏͳȀʹ
Ǥ
ሺ
ͳʹǤͻ
ɉɪɢɛɚɜɢɦ ɢ ɜɵɱɬɟɦ ɫɭɦɦɭ ɢɡ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜ ɭɪɚɜɧɟɧɢɢ (12.67), ɡɚɬɟɦ ɞɥɹ
ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɫɭɦɦɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ (12.68), ɚ ɞɥɹ ɫɥɚɝɚɟɦɨɝɨ ɫ ɦɢɧɭɫɨɦ ɩɨɥɨɠɢɦ ɜɟɪɯɧɢɣ ɪɚɜɧɵɦ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɩɨɥɭɱɢɦ
ஶ
ܶ
ൌ൬
ܶ
ݏͳȀʹͳȀͶߨܶ
௦ୀ
ɷɬɨɬ ɪɹɞ ɭɠɟ ɯɨɪɨɲɨ ɫɯɨɞɢɬɫɹ ɢ ɜɟɪɯɧɢɣ ɩɪɟɞɟɥ ɩɨɥɨɠɟɧ ɪɚɜɧɵɦ ɛɟɫɤɨɧɟɱɧɨɫɬɢ.
ɋɭɦɦɚ ɷɬɨɝɨ ɪɹɞɚ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɞɢɝɚɦɦɚ-ɮɭɧɤɰɢɸ ߰ሺݖሻൌሺȞሺݖሻሻ
ͳ
െ
߬
ͳ
ݏͳȀʹ
൰
ǡ
ሺ
ͳʹǤͲ
ᇱ
ɫ ɩɨɦɨ-
ɳɶɸ ɮɨɪɦɭɥɵ
ஶ
ͳ
߰ሺͳݖሻൌെߛ൬
ୀଵ
ͳ
െ
݊
݊ݖ
൰ǡ
ሺ
ͳʹǤͳ
ɜ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ
ܶ
ܶ
ɷɬɨ ɟɫɬɶ ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ܶ
ɫɥɭɱɚɣ ܶ
߬بͳ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɨɱɟɧɶ ɦɚɥɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɢɦɟɫɟɣ ɢɥɢ ɫɥɚ-
ɛɨɦɭ ɩɪɢɦɟɫɧɨɦɭ ɩɨɬɟɧɰɢɚɥɭ, ɬɨɝɞɚ ɦɵ ɦɨɠɟɦ ɪɚɡɥɨɠɢɬɶ ߰ሺͳȀʹݖ
ᇱ
ሺ
ɥɨɪɚ. ɍɱɢɬɵɜɚɹ, ɱɬɨ ߰
ͳȀʹሻൌߨ
ͳ
ൌ߰൬
ʹ
ଶ
Ȁʹ, ɩɨɥɭɱɢɦ
ܶ
ܶ
൰െ߰൬
ൌെ
ͺܶ
ͳ
ͳ
ʹ
ߨ
߬
൰ǡ
Ͷߨܶ
߬
. Ɋɚɫɫɦɨɬɪɢɦ ɩɪɟɞɟɥɶɧɵɣ
ǡ
ሺ
ͳʹǤʹ
ሻ
ɜ ɪɹɞ Ɍɟɣ-
ሺ
ͳʹǤ͵
ɨɬɤɭɞɚ
ܶ
ൌܶ
ߨ
െ
ǡ
ͺ߬
ሺ
ͳʹǤͶ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢɦɟɫɢ ɭɦɟɧɶɲɚɸɬ ɬɟɦɩɟɪɚɬɭɪɭ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɩɟɪɟɯɨɞɚ.
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
136
12.1.6. ɉɚɪɚɦɟɬɪ ɩɨɪɹɞɤɚ
ɉɪɢ ɧɢɡɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ܶ՜Ͳ ɜ ɭɪɚɜɧɟɧɢɢ (12.63) ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɨɬ ɫɭɦ-
ɦɢɪɨɜɚɧɢɹ ɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɸ ɩɨ ɩɪɚɜɢɥɭ
ஶ
ሺ
ʹߨܶ ՜න ݀߱ǡ
வ
ఠ
ೞ
ͳʹǤͷ
ɤɪɨɦɟ ɬɨɝɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɛɨɥɟɟ ɭɞɨɛɧɟɟ ɭɪɚɜɧɟɧɢɟ (12.63) ɦɨɞɢɮɢɰɢɪɨɜɚɬɶ ɞɪɭɝɢɦ ɫɩɨɫɨɛɨɦ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɫɨɨɬɧɨɲɟɧɢɹɦɢ (11.29)
ఌ
ʹɎݒ
ଶ
݃
ൌ
ʹߝ
ʹߝ
ி
ǡ
ȟ
ȁοȁ
ɬɨɝɞɚ ɩɪɨɞɟɥɵɜɚɹ ɬɪɸɤ ɫ ɩɪɢɛɚɜɥɟɧɢɟɦ ɢ ɜɵɱɢɬɚɧɢɟɦ ɪɚɜɟɧɫɬɜɚ
ஶ
ൌʹ߬න ݀߱ቊ
ͳ
߲
߲
ݖ
ܫ
߲ݖ
ଶఛఠ
ݖ
߲ݖ
ȁοȁ
ȟ
ɇɚɫ ɢɧɬɟɪɟɫɭɟɬ ɩɪɟɞɟɥ ߬ȁοȁ՜λ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɚɫɢɦɩɬɨɬɢɤɚɦɢ ܫ
ሺߥݖሻ
ɢ ɩɪɢ ߥ՜λ, ɫɩɪɚɜɨɱɧɢɤ ɞɚɟɬ ([14], ɫɬɪ. 199)
ܫ
ఔ
ܫ
ܫ
ᇱ
ఔ
ఔ
ሺߥݖሻ
ሺߥݖሻ
ൎ
ͳݖ
ξ
ଶ
ͳ
െ
ݖ
ʹߥݖ
ி
ൌන
ሺݖሻ
൨
௭ୀଶఛȁο
ʹߥݖሺͳݖ
ಷ
݀ߝ
ȁο
ǡ
ଶ
ȁ
ଶ
ඥ
ߝ
ሺ
ʹ߬ඥ߱
ሺߥ
ͳ
ଶ
ିଵ
ȁο
ሻ
ǡ
െ
ȁ
ͳ
ଶ
ሻ
ͳʹǤ
ቋ
Ǥ
ଶ
ȁ
ሺ
ͳʹǤ
ሻ
ሺͳʹǤ
ᇱ
ሺߥݖሻ
ɞɥɹ
ఔ
ሺ
ͳʹǤͺ
ɨɬɫɸɞɚ ɦɨɠɧɨ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ
ଶ
߱
ଶ
ଶ
ሺ
߱
ȁο
ଶሻଶ
ȁ
ǡሺͳʹǤͻ
ͳ
߲
߲
ݖ
ݖ
߲ݖ
߲ݖ
ܫ
ଶఛఠ
ሺݖሻ
൨
௭ୀଶȁఊ
ൌ
ȁ
ʹ߬ඥ߱
ͳ
ଶ
ȁο
െ
ଶ
Ͷ߬
ȁ
ሻ
ሻ
ሻ
ሻ
ሻ
ɩɨɞɫɬɚɜɥɹɹ ɜ (12.77) ɢ ɜɵɱɢɫɥɹɹ ɢɧɬɟɝɪɚɥ, ɩɨɥɭɱɢɦ
ɷɬɨ ɞɚɟɬ ɩɪɢ ߬ȟ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɳɟɥɶ
ɤɪɢɬɢɱɟɫɤɨɦ ߬
بͳ
ȁοȁ
ɨɧɚ ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɨɣ ɧɭɥɸ, ɬɨ ɟɫɬɶ ɩɪɢɦɟɫɢ ɩɨɥɧɨɫɬɶɸ ɩɨɞɚɜ-
ɥɹɸɬ ɩɟɪɟɯɨɞ. ɑɬɨɛɵ ɧɚɣɬɢ ߬
ȁοȁ
ȟ
ȁοȁ
ൌȟ
ɭɦɟɧɶɲɚɟɬɫɹ ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ߬ ɢ ɩɪɢ ɧɟɤɨɬɨɪɨɦ
, ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɚɫɢɦɩɬɨɬɢɤɨɣ (12.66) ɩɪɢ
ൌെ
ߨ
ͺȁοȁ߬
ߨ
െ
ͺ߬
ǡ
Ǥ
ɢ ɩɨɞɫɬɚɜɢɦ ɜ (12.77)
ஶ
ൌʹ߬න ݀߱
ͳ
ቊ
ʹ߱߬ͳ
െ
ʹ߬ඥ߱ଶȁο
ͳ
െ
ଶ
ሺ
ȁ
ȁοȁ
ȟ
137
ʹሺȁοȁ߬
ʹ߱߬ͳ
ሻଶሺ
ʹ߱߬ʹ
ሺ
ሻ
ͳʹǤͺͲ
ሺ
ሻ
ͳʹǤͺͳ
ȁοȁ
՜Ͳ
ଶ
ሻ
ሺ
ͳʹǤͺʹ
ሻ
Ǥ
ቋ
ሻ
ɉɟɪɜɵɟ ɞɜɚ ɫɥɚɝɚɟɦɵɟ ɞɚɸɬ ȁοȁ߬, ɚ ɜɨ ɜɬɨɪɨɦ ɩɟɪɟɯɨɞɢɦ ɤ ɩɟɪɟɦɟɧɧɨɣ
ݔൌʹ߱߬, ɩɨɥɭɱɢɦ
ȟ
߬ൌʹ
߬
න
ଶ
ሺ
ሻ
ȁοȁ
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɩɪɢ ߬
ȁοȁ
ஶ
ሺ
ݔͳ
ିଵ
ൌȟ
ൎͳǡʹͺȟ
݀ݔ
ଶ
ሻ
ሺ
ݔʹ
ȁοȁ
,
ɢɫɱɟɡɚɟɬ, ɚ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ ɬɨɱɤɢ
ሾ
߬Ȁ߬
ൌʹ
ሻ
െͳ
ሿ
ሺ
ଵȀଶ
ȁοȁ
߬
ሻଶሺ
ͳെʹሻǤ
ሺ
ͳʹǤͺ͵
ሺ
ͳʹǤͺͶ
12.2. ȼɥɢɹɧɢɟ ɩɪɢɦɟɫɟɣ ɧɚ ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ
ɉɪɢɦɟɧɢɦ ɪɚɡɜɢɬɭɸ ɬɟɯɧɢɤɭ ɤ ɢɫɫɥɟɞɨɜɚɧɢɸ ɩɨɜɟɞɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɫɨɫɬɨɹɧɢɣ ɩɚɣɟɪɥɫɨɜɫɤɨɝɨ ɞɢɷɥɟɤɬɪɢɤɚ ɜ ɩɪɢɫɭɬɫɬɜɢɢ ɫɥɭɱɚɣɧɨɝɨ ɩɨɬɟɧɰɢɚɥɚ ɩɪɢɦɟɫɟɣ. Ȼɨɥɶɲɨɣ ɢɧɬɟɪɟɫ ɩɪɢ ɷɬɨɦ ɩɪɟɞɫɬɚɜɥɹɟɬ ɜɨɩɪɨɫ ɨ ɫɭɳɟɫɬɜɨɜɚɧɢɢ ɡɚɩɪɟɳɟɧɧɨɣ ɳɟɥɢ. Ⱦɥɹ ɜɵɱɢɫɥɟɧɢɹ ɩɥɨɬɧɨɫɬɢ ɫɨɫɬɨɹɧɢɣ ɜɨɫɩɨɥɶɡɭɟɦɫɹ
ɮɨɪɦɭɥɨɣ (11.35), ɜ ɤɨɬɨɪɭɸ ɩɨɞɫɬɚɜɢɦܩ
Ȁ௩
బ
නሾ݃ො
ߥሺ߱ሻൌ
ͳ
ߨܮ
ɉɨɞɫɬɚɜɥɹɟɦ ɜ ɷɬɭ ɮɨɪɦɭɥɭ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ݃ො
ɷɥɟɦɟɧɬɵ S-ɦɚɬɪɢɰɵ ɫɨɝɥɚɫɧɨ (8.67), ɩɨɥɭɱɢɦ
Ȁ௩
బ
ൌ
ͳ
න ቈെ
ߨܮ
መ
ሺ
ܵ
ݐǡെλሻԜܵ
ఠԜଵଶ
መ
ܵ
ఠԜଶଶ
መ
ሺ
λǡെλ
ఠԜଶଵ
ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɪɚɡɥɨɠɟɧɢɟɦ (12.13) ɞɥɹ ൣܵ
Ȁ௩
బ
න ݀ݐ
ߥሺ߱ሻൌ
ͳ
ߨܮ
ɝɞɟ ܽሺݐሻ ɢ ܾሺݐሻ ɞɚɸɬɫɹ (12.15). ɍɫɪɟɞɧɹɟɦ ɩɨ ɫɥɭɱɚɣɧɵɦ ɩɨɥɹɦ, ɩɨɥɭɱɚɟɦ
ͳ
ͳʹሺെͳ
ߨݒ
ͳʹሺെͳ
ɝɞɟ ܣ
ߥሺ߱ሻൌ
ɢ ܤ
ɞɚɸɬɫɹ (12.30), ɜ ɢɬɨɝɟ ɩɨɥɭɱɚɟɦ
ߥሺ߱ሻൌ
ͳ
ߨݒ
ோ
ሺ
߱ǡǡ
ோ
ሺ
߱ǡݐǡݐሻെ݃ො
ଵଵ
ߥሺ߱ሻൌ
ሺ
ሻ
λǡݐ
መ
ሻ
ܵ
ఠԜଶଶ
መ
ఠԜଶଶ
ஶ
ͳʹሺെͳ
ୀଵ
ஶ
ୀଵ
ஶ
ሻ
ȁߛȁ
ୀଵ
ሻ
ɜ ɜɢɞɟ (8.35), ɩɨɥɭɱɢɦ
ோ
ሺ
ሻ
ሿ
߱ǡݐǡݐ
ଶଶ
ோ
ሺ
߱ǡݐǡݐሻ ɢ ݃ො
ଵଵ
ሺ
ݐǡെλሻԜܵ
መ
ሺ
ܵ
ఠԜଶଶ
ሺ
λǡെλሻ൧
ܽ
ሻ
ሻ
ܣ
ଶ
ଶ
ܳ
ଵ
መ
ఠԜଶଶ
λǡെλ
ሺݐሻ
ܾ
ܤ൩ǡ
ଶ
ܳ
ǥܳ
ଶ
݀ݐǤ
ோ
ଶଶ
ሺ
λǡݐ
ሻ
ିଵ
, ɩɨɥɭɱɢɦ
ሺݐሻ
൩ǡ
ଶ
൩ǡ
ሻ
ሺ
߱ǡݐǡݐ
݀ݐǡ
ሺ
ͳʹǤͺͷ
ሻ
ɱɟɪɟɡ
ሺͳʹǤͺ
ሺ
ͳʹǤͺ
ሺ
ͳʹǤͺͺ
ሺ
ͳʹǤͺͻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
138
ଶ
ɡɞɟɫɶ ܳ
ሺെ݅ߚǡȁߛ
ɦɵ ɩɪɨɜɟɥɢ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɩɪɨɞɨɥɠɟɧɢɟ ɫ ɦɧɢɦɵɯ ɞɢɫɤɪɟɬɧɵɯ ɱɚɫɬɨɬ ݅߱
ɞɟɣɫɬɜɢɬɟɥɶɧɭɸ ɨɫɶ ߱ ɩɨ ɩɪɚɜɢɥɭ ߱
ɥɨɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɹ ɞɪɨɛɢ, ɫɭɦɦɭ ɜ (12.89) ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ߲ܳ
ȁ
ሻ – ɧɟɩɪɟɪɵɜɧɚɹ ɞɪɨɛɶ (12.45) ɫ ߚൌʹ߬߱ ɢ ߛൌ߬ο, ɩɪɢ ɷɬɨɦ
՜െ݅߱ (ɢɥɢ ߚ௦՜െ݅ߚ). ɉɨɥɶɡɭɹɫɶ ɩɪɚɜɢ-
௦
ଵ
ɧɚ
௦
Ȁ߲ߚ,
ɷɬɨ ɞɚɟɬ
ߥሺ߱ሻൌ
ͳ
ቈͳ݅ʹȁߛ
ߨݒ
ሺ
െ݅ߚǡȁߛ
߲ܳ
ଵ
ଶ
ȁ
߲ߚ
ଶ
ሻ
ȁ
ǡ
ሺ
ͳʹǤͻͲ
ɞɚɥɟɟ ɫɨɝɥɚɫɧɨ (12.60), ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ ɱɟɪɟɡ ɨɬɧɨɲɟɧɢɟ ɮɭɧɤɰɢɣ Ȼɟɫɫɟɥɹ
ሺ
ሻ
ߥሺ߱ሻൌ
ͳ
ቈͳ݅ʹȁߛȁ
ߨݒ
߲
߲ߚ
ܫ
ଵିఉ
ܫ
ିఉ
ሺ
ʹȁߛ
ʹȁߛ
ȁ
Ǥ
ሻ
ȁ
ሺ
ͳʹǤͻͳ
Ⱦɚɥɟɟ ɭɱɬɟɦ ɫɨɨɬɧɨɲɟɧɢɟ
ܫ
ଵିఉ
ሺݖሻ
ൌܫ
ᇱ
ିఉ
ሺݖሻ
݅
ߚ
ሺݖሻ
ିఉ
ǡ
ܫ
ݖ
ሺ
ͳʹǤͻʹ
ɷɬɨ ɩɪɢɜɨɞɢɬ ɤ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɭɥɟ
ሻ
ሻ
ሻ
ߥሺ߱ሻൌെ
ȁοȁ
ߨݒ
߲
߲߱
߲
൭
ܫ
߲ݖ
ିଶఠఛ
ሺݖሻ
൱
௭ୀଶఛȁο
Ǥ
ȁ
ሺ
ͳʹǤͻ͵
Ⱦɥɹ ɪɚɡɴɹɫɧɟɧɢɹ ɜɨɩɪɨɫɚ ɨ ɫɭɳɟɫɬɜɨɜɚɧɢɢ ɡɚɩɪɟɳɟɧɧɨɣ ɳɟɥɢ ɜɵɱɢɫɥɢɦ ɜɟ-
ɥɢɱɢɧɭ ߥሺͲሻ. Ⱦɥɹ ɷɬɨɝɨ ɮɭɧɤɰɢɸ Ȼɟɫɫɟɥɹ ɜ (12.93) ɪɚɡɥɨɠɢɦ ɜ ɪɹɞ ɩɨ ߚ, ɷɬɨ ɞɚɟɬ
ሺݖሻ
߲ܫ
ܫ
ିଶఠఛ
ሺݖሻ
ൌܫ
ሺݖሻ
െ݅ʹ߱߬ቆ
ఔ
߲ߥ
ቇ
ఔୀ
ൌܫ
ሺݖሻ
݅ʹ߱߬ܭ
ሺݖሻǡሺ
ͳʹǤͻͶ
ɝɞɟ ɦɵ ɜɨɫɩɨɥɶɡɨɜɚɥɢɫɶ ɮɨɪɦɭɥɨɣ ([14], ɫɬɪ. 198)
ሺݖሻ
߲ܫ
ఔ
߲ߥ
ቇ
ఔୀ
ቆ
ൌെܭ
ሺݖሻ
ǡ
ሺ
ͳʹǤͻͷ
ɞɚɥɟɟ, ɪɚɫɤɥɚɞɵɜɚɟɦ ɜ ɪɹɞ ɩɨ ߱߬, ɩɨɥɭɱɢɦ
ሺݖሻ
ܭ
ܫ
ିଶఠఛ
ሺݖሻ
ൌܫ
ሺݖሻ
݅ʹ߱߬
ǡ
ሺݖሻ
ܫ
ሺ
ͳʹǤͻ
ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɧɟ ɫɨɞɟɪɠɢɬ ߱ ɢ ɨɧɨ ɢɫɱɟɡɚɟɬ ɩɪɢ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɢ, ɞɚɥɟɟ
ߥሺͲሻൌെ
ʹȁοȁ߬
ߨݒ
ɬɟɩɟɪɶ ɭɱɬɟɦ ܭ
ᇱ
߲
ܭ
ቆ
߲ݖ
ܫ
ሺݖሻ
ൌെܭ
ߥሺͲሻൌ
ሺݖሻ
ሺݖሻ
ቇ
௭ୀଶఛȁο
ሺݖሻ
ଵ
ʹȁοȁ߬
ߨݒ
ᇱ
ሺݖሻ
ܫ
ଶ
ሺݖሻ
ሺݖሻ
ܫ
ቈ
, ɬɨɝɞɚ
ሺݖሻ
ଵ
ܭ
ሺݖሻ
ܭ
ʹȁοȁ߬
ൌെ
ߨݒ
ȁ
ᇱ
ሺݖሻ
, ܫ
ൌܫ
ሺݖሻ
ܫ
ቈ
ଵ
ሺݖሻ
ܭ
ଵ
ܫ
ሺݖሻ
ܫ
ᇱ
െܫ
ଶ
ሺݖሻ
௭ୀଶȁοȁఛ
ሺݖሻ
Ǥ
ܭ
ሺݖሻ
ǡሺͳʹǤͻ
ሺ
ͳʹǤͻͺ
139
ሻ
ሻ
ሻ
ሻ
ሻ
ሻ
ȼɵɪɚɠɟɧɢɟ ɜ ɱɢɫɥɢɬɟɥɟ ɜ ɤɜɚɞɪɚɬɧɵɯ ɫɤɨɛɤɚɯ ɟɫɬɶ ɜɪɨɧɫɤɢɚɧ
ሺݖሻ
ሺݖሻ
ܭ
ܫ
ଵ
ܫ
ଵ
ሺݖሻ
ܭ
ሺݖሻ
ͳ
ൌ
ǡ
ݖ
ሺ
ͳʹǤͻͻ
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɭɱɚɟɦ ɨɤɨɧɱɚɬɟɥɶɧɭɸ ɮɨɪɦɭɥɭ
ሻ
ߥሺͲ
ߥ
ൌͳȀߨݒ
ɝɞɟ ߥ
ɥɢɱɢɧɚ ߥሺͲ
ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ ɦɟɬɚɥɥɚ ɛɟɡ ɩɪɢɦɟɫɟɣ. Ɇɵ ɩɨɥɭɱɚɟɦ, ɱɬɨ ɜɟ-
ሻ
ɧɟ ɪɚɜɧɚ ɧɭɥɸ ɩɪɢ ɥɸɛɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɢɦɟɫɟɣ. ȿɫɥɢ
ɦɵ ɩɨɥɭɱɚɟɦ ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ ɱɢɫɬɨɝɨ ɦɟɬɚɥɥɚ ߥ
ȁȟȁ
ɱɚɟ
߬بͳ, ɢɫɩɨɥɶɡɭɹ ɚɫɢɦɩɬɨɬɢɤɭ ܫ
ߥሺͲ
ߥ
ିଶ
ൌܫ
ሻ
ൎͶߨȁȟȁ߬݁
ሺ
ʹȁȟȁ߬ሻǡ
ሺݖሻ
݁
. ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭ-
௭
Ȁξʹߨݖǡ ɩɨɥɭɱɚɟɦ
ିଶȁȁఛ
Ǥ
ሺ
ͳʹǤͳͲͲ
ȁȟȁ
߬՜Ͳ, ɬɨ
ሺ
ͳʹǤͳͲͳ
Ɋɟɡɭɥɶɬɚɬ (12.100) ɨɡɧɚɱɚɟɬ, ɱɬɨ ɩɪɢ ɥɸɛɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɢɦɟɫɟɣ ɷɧɟɪɝɟ-
ɬɢɱɟɫɤɚɹ ɳɟɥɶ ɢɫɱɟɡɚɟɬ, ɯɨɬɹ ɩɪɢ ɷɬɨɦ ɩɥɨɬɧɨɫɬɶ ɫɨɫɬɨɹɧɢɣ ɩɪɢ ߱ൌͲɹɜɥɹɟɬɫɹ
ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ ɦɚɥɨɣ ɜɟɥɢɱɢɧɨɣ ɩɪɢ ɦɚɥɵɯ ɤɨɧɰɟɧɬɪɚɰɢɹɯ ɩɪɢɦɟɫɟɣ.
ɋɥɟɞɭɟɬ ɩɨɞɱɟɪɤɧɭɬɶ, ɱɬɨ ɨɬɫɭɬɫɬɜɢɟ ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɳɟɥɢ ɫɨɜɫɟɦ ɧɟ ɨɡɧɚ-
ɱɚɟɬ ɧɚɥɢɱɢɟ ɤɨɧɟɱɧɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɩɪɢ ܶൌͲ.
Ⱥɧɚɥɨɝɢɱɧɨ ɫɢɬɭɚɰɢɢ ɫ ɧɨɪɦɚɥɶɧɵɦ ɨɞɧɨɦɟɪɧɵɦ ɦɟɬɚɥɥɨɦ ɡɞɟɫɶ
ɫɭɳɟɫɬɜɭɟɬ ɤɜɚɡɢɧɟɩɪɟɪɵɜɧɵɣ ɫɩɟɤɬɪ ɥɨɤɚɥɢɡɨɜɚɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɩɪɢɪɨɞɚ ɤɨɬɨɪɵɯ
ɦɨɠɟɬ ɛɵɬɶ ɭɫɬɚɧɨɜɥɟɧɚ ɜɵɱɢɫɥɟɧɢɟɦ ɩɪɨɜɨɞɢɦɨɫɬɢ. ɑɬɨ ɛɭɞɟɬ ɩɪɨɞɟɥɚɧɨ ɜ
ɫɥɟɞɭɸɳɟɣ ɝɥɚɜɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɜ ɡɚɤɥɸɱɟɧɢɟ ɩɪɟɞɟɥ ɨɱɟɧɶ ɫɥɚɛɨɝɨ ɪɚɫɫɟɹɧɢɹ ɧɚ ɩɪɢɦɟɫɹɯ. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ߬߱بͳǡ߬ȁοȁبͳ, ɢ ɩɨɷɬɨɦɭ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɚɫɢɦɩɬɨɬɢɤɭ
(12.78), ɤɨɬɨɪɚɹ ɞɚɟɬ
ሻ
ሻ
ሻ
൭
߲ݖ
ଶ
ඥ
߲
ܫ
ିଶఠఛ
ሺݖሻ
൱
௭ୀଶఛȁο
ൎെ݅
ȁ
߱
െȁο
ȁοȁ
ଶ
ȁ
ͳ
െ
ʹ߬ȁο
ǡ
ȁ
ɩɨɞɫɬɚɧɨɜɤɚ ɜ (12.93) ɞɚɟɬ ɪɟɡɭɥɶɬɚɬ ɞɥɹ ɱɢɫɬɨɝɨ ɦɟɬɚɥɥɚ (11.58)
ߥሺͲ
ߥ
ሻ
ቊ
ൌ
ȁ߱ȁ
Ͳ˒˓ˋȁ߱ȁ൏ȁȟȁ
ଶ
ଶ
െȁȟ
ȁ
ሺ߱
ଵ
ି
ଶ
ሻ
˒˓ˋȁ߱ȁȁȟȁ
ǡ
ɩɪɢɱɟɦ ɷɬɚ ɮɨɪɦɭɥɚ ɩɨɥɭɱɟɧɚ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɫɥɚɝɚɟɦɵɯ ɩɨɪɹɞɤɚ ሺ߱߬ሻ
Ʌɢɬɟɪɚɬɭɪɚ
ɋɨɞɟɪɠɚɧɢɟ ɷɬɨɣ ɝɥɚɜɵ ɨɫɧɨɜɚɧɨ ɧɚ ɦɚɬɟɪɢɚɥɟ ɪɚɛɨɬɵ [8].
140
ሺ
ͳʹǤͳͲʹ
ሺ
ͳʹǤͳͲ͵ሻ
ିଶ
.
ሻ
Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]